Does married filing jointly mean we also file our FBAR jointly?

The IRS says income-tax filing status, married filing jointly or separately, has no effect on the spousal exception. A non-U.S.-person spouse who joins a joint return through an election under IRC 6013(g) or (h) doesn't become a U.S. person for the FBAR. Start with each spouse's foreign accounts and signature authority, then check whether every account the non-filing spouse must report is jointly owned with the filing spouse.⁠^{[1]Source 1}⁠^{[2]Source 2}⁠^{[3]Source 3}

A couple files Form 1040 jointly. One spouse also holds a foreign savings account in their own name, in addition to the couple's joint accounts. That spouse can't be the non-filing spouse. The couple can still file one FBAR if the spouse with the separate account is the one who files.⁠^{[3]Source 3}

Apply the FBAR rules to each spouse, whatever your tax filing status. If both spouses must file separately, each reports the full value of every jointly owned account.⁠^{[2]Source 2}⁠^{[4]Source 4}